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• W2040990925 abstract "Let $xi_{0},xi_{1},ldots$ be independent identically distributed complex-valued random variables such that $mathbb{E}log(1+|xi_{0}|)<infty$. We consider random analytic functions of the form [mathbf{G}_{n}(z)=sum_{k=0}^{infty}xi_{k}f_{k,n}z^{k},] where $f_{k,n}$ are deterministic complex coefficients. Let $mu_{n}$ be the random measure counting the complex zeros of $mathbf{G}_{n}$ according to their multiplicities. Assuming essentially that $-frac{1}{n}log f_{[tn],n}to u(t)$ as $ntoinfty$, where $u(t)$ is some function, we show that the measure $frac{1}{n}mu_{n}$ converges in probability to some deterministic measure $mu$ which is characterized in terms of the Legendre–Fenchel transform of $u$. The limiting measure $mu$ does not depend on the distribution of the $xi_{k}$’s. This result is applied to several ensembles of random analytic functions including the ensembles corresponding to the three two-dimensional geometries of constant curvature. As another application, we prove a random polynomial analogue of the circular law for random matrices." @default.
• W2040990925 created "2016-06-24" @default.
• W2040990925 creator A5010851411 @default.
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• W2040990925 date "2014-07-01" @default.
• W2040990925 modified "2023-10-16" @default.
• W2040990925 title "Asymptotic distribution of complex zeros of random analytic functions" @default.
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• W2040990925 doi "https://doi.org/10.1214/13-aop847" @default.
• W2040990925 hasPublicationYear "2014" @default.
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